Varieties of Modality

Modal statements tell us something about what could be or must be
the case. Such claims can come in many forms. Consider:

No one can be both a bachelor and
married. (‘Bachelor’ means ‘unmarried
man’.)

You could not have been born of different
parents. (Someone born of different parents wouldn't be you.)

Nothing can travel faster than light.
(It's a law of nature.)

One cannot get from London to New York in
less than one hour. (Planes that fast haven't been developed
yet.)

You cannot leave the palace. (The doors
are locked.)

You cannot promise to come and then stay
at home. (It's just wrong.)

You cannot start a job application cover
letter with “hey guys”. (It's just not done.)

You cannot castle if your king is in
check. (It's against the rules.)

You cannot deduct your holidays from your
taxes. (It's against the law.)

Fred cannot be the killer. (The evidence shows that
he's innocent.)

Each of these claims appears to have a true reading. But it also
seems that ‘cannot’ needs to be interpreted in different
ways to make the different sentences true. For one thing, we can, in
the same breath, accept a modal claim in one of the senses illustrated
by (1)–(10) while rejecting it in another one of these senses,
as in the following dialogue:

Caesar:

You're lucky that I'm still
here. The doors were unlocked. I could have left the palace.

Cleopatra:

True. But then again, you couldn't have left the
palace. That would have been wrong, given that you promised to meet me
here.

Moreover, the modal claims (1)–(10) appear to be true for
completely different reasons. For example, it may be held that the
truth of (1) is due to the meanings of its constituent expressions;
that (2) holds because it lies in your nature to be born of your actual
parents; that (3) is true because the laws of nature preclude
superluminal motion; that (4) holds because of technological
limitations; that (5) owes its truth to the presence of insurmountable
practical obstacles; that (6)–(9) are made true by the demands
of morality, etiquette, the rules of chess, and the law respectively;
and that (10) holds because the known facts prove Fred's
innocence.

It is one of the tasks of a philosophical theory of modality to give
a systematic and unified account of this multiplicity of modal
concepts. This article discusses a few of the main issues that need to
be addressed by anyone pursuing this goal. Sections 1 and 2 concern the
question of what fundamental categories of modal notions there are. The
focus will be on two contemporary debates: whether there are separate
forms of modality that are tied to the epistemic and the metaphysical
domains (section 1), and whether there is a special kind of necessity
associated with the laws of nature (section 2). Section 3 discusses
questions about the relations between different notions of necessity.
Can some of them be reduced to other, more fundamental ones? If so,
which concepts of necessity are the most fundamental ones? And if there
are several fundamental kinds of necessity, what do they have in common
that makes them all kinds of necessity?

There are many ways the world could have been. You could
have gotten up later today. Your parents could have failed to meet, so
that you were never born. Life could never have developed on
earth. The history of the universe could even have been completely
different from the beginning. And many philosophers believe that the
laws of nature could have been different as well (although that has
been denied, as discussed in section 2). Maximally specific ways the
world could have been are commonly called ‘possible
worlds.’ The apparatus of possible worlds allows us to introduce
a set of modal notions: a proposition is necessary just in case it is
true in all possible worlds, a proposition is possible just in case it
is true in some possible worlds, and it is contingent just in case it
is true in some but not all possible worlds. A sentence is
necessary (possible, contingent) just in case it expresses a necessary
(possible, contingent) proposition.

The modal notions considered in the last paragraph are not obviously
epistemological. On the face it, we are not reporting a fact about what
is or can be known or believed by anyone when we say that life could
have failed to develop. But there is also a family of modal concepts
that are clearly epistemological. These are the notions we employ when
we say things like ‘Fred must have stolen the book (the
evidence shows conclusively that he did it),’ or ‘Mary
cannot be in London (she would have called me).’ These
modal utterances seem to make claims about what the available evidence
shows, or about which scenarios can be ruled out on the basis of the
evidence. More formally, we can say that a proposition P is
epistemically necessary for an agent A just in case the
empirical evidence A possesses and ideal reasoning (i.e.,
reasoning unrestricted by cognitive limitations) are sufficient to
rule out ~P. This notion of epistemic necessity is
agent-relative: one and the same claim can be epistemically necessary
for one agent, but not for another agent with less empirical
evidence. We obtain a notion of epistemic necessity of particular
philosophical interest by focusing on a limiting case, namely that of
a possible agent with no empirical evidence
whatsoever.[1] A
proposition P is epistemically necessary for such an agent
just in case ideal reasoning alone, unaided by empirical evidence, is
sufficient to rule out ~P. A proposition that meets this
condition can be called a priori in at least one sense of
this term, or we can call it simply epistemically necessary
(without relativization to an agent). Propositions that are not a
priori are called a
posteriori.[2]

It is an important and controversial question whether the necessary
propositions are all and only the epistemically necessary (a priori)
ones, or whether the extensions of the two concepts can come apart. One
possible reason for thinking that the notions are coextensive derives
from a very natural picture of information and inquiry. On this
picture, all information about the world is information about which of
all possible worlds is realized (i.e., about where in the space of all
possible worlds the actual world is located). My total information
about the world can be identified with the set of possible worlds that
I cannot rule out on the basis of my empirical evidence and ideal
reasoning. As I gather more and more empirical evidence, I can
progressively narrow down the range of possibilities. Suppose, for
example, that I am ignorant of the current weather conditions. The
worlds compatible with my evidence include some where the weather is
good and others where it is bad. A look out of the window at the rain
provides information about the matter. I can now narrow down the set of
possibilities by excluding all possible worlds with fine weather. On
this account, a proposition P is epistemically necessary
for A just in case P is true in all possible worlds that
cannot be ruled out on the basis of A's empirical
evidence and ideal reasoning. P is a priori just in case it is
epistemically necessary for a possible agent who has no empirical
evidence. Since such an agent cannot rule out any possible worlds, a
proposition is a priori just in case it is true in all possible worlds.
In other words, the a priori propositions are all and only the
necessary
propositions.[3]

This approach is often combined with a certain account of semantic
content. One of the main purposes of language is to transmit
information about the world. Where P is any sentence used for
that purpose (roughly speaking, a declarative sentence), it seems
natural to think of P's content (the proposition
expressed by it) as the information that is semantically encoded in it.
Combining this with the foregoing account of information, we can think
of the content of a sentence as a set of possible worlds (namely, the
set containing just those worlds of which the sentence is true) or,
equivalently, as a function from worlds to truth-values.

This picture connects the modal, epistemic and semantic realms in a
simple and elegant way, and various versions of it have informed the
work of numerous contemporary philosophers (including David Lewis,
Robert Stalnaker, David Chalmers, and Frank Jackson). However, the
approach has come under pressure from data to be considered in the next
section.

The idea that all and only the a priori truths are necessary
was thrown into serious doubt by the work of philosophers including
Hilary Putnam (1972) and Saul Kripke (1980). Kripke distinguishes
between two different kinds of singular terms, rigid
and non-rigid ones. A so-called rigid designator is
an expression that singles out the same thing in all possible
worlds. Kripke argues that ordinary proper names like ‘Al
Gore’ are rigid. We can use this name to describe how things
actually are, e.g., by saying ‘Al Gore became vice president in
1993.’ In such cases, the name picks out Al Gore. But we can
equally use the name to describe how things stand in other possible
worlds, e.g., by saying, ‘If Bill Clinton had chosen a different
running mate, Al Gore would not have become vice president.’ In
this case, we are talking about a non-actualized possibility, and we
use the name ‘Al Gore’ to describe this
possibility. Moreover, we use the name to say something about how
things stand with Al Gore in that possibility. In general, when we use
the name to describe any possible world, we use it to talk about the
same person, Al Gore. Other examples of rigid designators include
indexical expressions like the first-person pronoun ‘I,’
or the expression ‘now.’ When you use the term
‘I’ to describe any possible world, you are always picking
out the same thing: yourself. Natural kind terms like
‘water’ and ‘gold’ can also be regarded as
rigid terms, as they single out the same kinds in every possible
world. Non-rigid singular term, by contrast, pick out different
entities in different possible scenarios. The paradigmatic examples of
non-rigid terms are descriptions that are satisfied by different
objects in different possible worlds. For example, ‘the most
annoying person in the history of the world’ may pick out Fred
in the actual world, while picking out Cleopatra in some other
possible worlds.

Singular terms can be introduced into the language with the help of
descriptions. There are two ways in which that can be done. On the one
hand, we can stipulate that the singular term is to be
synonymous with the description, for example by laying down
that ‘the morning star’ is to mean the same as ‘the
last celestial body to be seen in the morning.’ When we use the
expression to describe another possible world, the new expression will
single out whatever celestial body is the last one that can be seen in
the morning in that world. Since different things meet this condition
in different worlds, the expression is non-rigid. On the other hand, we
may introduce a term with the stipulation that it is to be a rigid
designator referring to whatever object actually satisfies the
description. For instance, we may lay it down that
‘Phosphorus’ is to refer rigidly to the object that is
actually the last celestial body visible in the morning. Since that
object is Venus, the name will pick out Venus, not only when we use it
to describe the actual world, but also when we (in the actual world)
use it to describe other possible worlds, including worlds where Venus
is not the last planet visible in the morning.When a description is
used to introduce a singular term in the second way, it merely serves
to fix the reference of the term, but is not synonymous with it.

Now consider a true identity statement that involves two rigid
designators, such as

(1)

Mark Twain (if he exists) is Samuel Clemens.

Since ‘Mark Twain’ and ‘Samuel Clemens’ pick
out the same entity in every possible world where they pick out
anything, this identity statement is a necessary truth. (Note that the
statement is conditionalized on Mark Twain's existence, which
makes it possible to avoid the question whether (1) is true in worlds
where the two names pick out nothing.) But it is far from immediately
obvious that (1) expresses something that can be known a priori. At
least on the face of it, we may think that someone who knows her
neighbor by the name of ‘Samuel Clemens,’ who has read
several stories by an author named ‘Mark Twain’ and who
fails to realize that her neighbor and the author are identical may not
know that which is expressed by (1). Moreover, it may seem that her
ignorance is irremediable by reasoning alone, that she requires
empirical evidence to come to know that which is stated by (1).

Another type of apparent counterexample to the thesis that all and
only the a priori truths are necessary concerns sentences like

(2)

If gold exists, then it has atomic number 79.

It seems plausible that it is an essential property of gold to have
atomic number 79: gold could not have (existed but) failed to have that
property. (A substance in another possible world that fails to have
atomic number 79 simply isn't gold, no matter how similar it may
otherwise be to the gold of the actual world.) And yet it seems clear
that it can only be known empirically that gold has that atomic number.
So, while (2) is a necessary truth, what it says cannot be known a
priori. For another illustration of this phenomenon, suppose that I
point to the wooden desk in my office and say:

(3)

If this desk exists, it is made of wood.

It is arguably essential to this desk to be made of wood. A desk in
another possible world that isn't wooden simply can't be
this desk, no matter how similar it may otherwise be to my
desk. But it seems that we need empirical evidence to know that the
desk is made of wood. So, (3) is another apparent example of a
necessary a posteriori truth.

Just as Kripke claims that some truths are necessary without being a
priori, he argues that a truth can be a priori without being necessary.
To use an example of Gareth Evans's (1982), suppose that I
introduce the term ‘Julius’ by stipulating that it is to
refer rigidly to the person who is in fact the inventor of the zip (if
such a person exists). Then it may appear that I don't need
further empirical evidence to know that

(4)

If Julius exists, then Julius is the inventor of the zip.

But (4) does not seem to be a necessary truth. After all, Julius
could have become a salesperson rather than an inventor.

According to Kripke, our initial surprise at the divergent
extensions of a prioricity and necessity should be mitigated on
reflection. A prioricity (epistemic necessity) is an
epistemological notion: it has to do with what can be known.
That is not true of the concept of necessity. (2) is necessary because
the atomic number of gold is an essential feature of it, and on the
face of it, that has nothing to do with what is known or believed by
anyone. This kind of necessity is a metaphysical notion, and
we may use the term ‘metaphysical necessity’ to distinguish
it more clearly from epistemic necessity.

Kripke's examples are not the only ones that could be appealed
to in order to shed doubt on the coextensiveness of necessity and a
prioricity. Some other problematic cases are listed below (Chalmers
2002a).

Mathematical truths. It is common to hold that all
mathematical truths are necessary. But on the face of it, there is no
guarantee that all mathematical truths are knowable a priori (or
knowable in any way at all). For example, either the continuum
hypothesis or its negation is true, and whichever of these claims is
true is also necessary. But for all we know, there is no way for us to
know that that proposition is true.

Laws of nature. Some necessitarians about the natural
laws (see section 2) believe that the laws hold in all metaphysically
possible worlds. But they are not a priori truths.

Metaphysical principles. It is often believed that
many metaphysical theses are necessary if true, e.g., theses about the
nature of properties (e.g., about whether they are universals, sets or
tropes) or the principle of unrestricted mereological composition. But
it is not obvious that all truths of this kind are a priori.

Principles linking the physical and the mental. Some
philosophers hold that all truths about the mental are metaphysically
necessitated by the physical truths, but deny that it is possible to
derive the mental truths from the physical ones by a priori reasoning
(see Hill & McLaughlin 1999; Yablo 1999; Loar 1999; and Chalmers
1999 for discussion). On that account, some of the conditionals that
link physical and mental claims are metaphysically necessary but not a
priori.

These examples are controversial. For any given mathematical claim
whose truth-value is unknown, one could hold that it is only our
cognitive limitations that have prevented us from establishing or
refuting the statement, and that the question could be decided by ideal
reasoning (so that the truth of the matter is a priori). Alternatively,
it may be held that the truth-value of the mathematical statement is
indeterminate. (Perhaps our practices do not completely determine the
references of all the terms used in the mathematical claim). The same
two options are available in the case of metaphysical principles.
Alternatively, one may argue that the relevant metaphysical theses are
merely contingent (see, e.g., Cameron 2007). Necessitarianism about the
natural laws is highly controversial and may simply be denied. And in
response to (iv), one may deny that the physical truths metaphysically
necessitate the mental truths (Chalmers 1996), or one may hold that the
mental truths can be derived from the physical ones by a priori
reasoning (Jackson 1998).

Philosophers have paid more attention to the examples given by
Kripke than to other possible cases of the necessary a posteriori, and
for that reason the discussion in the rest of this section will mostly
focus on Kripke's cases. Two strategies for explaining these
examples can be distinguished. Dualists about metaphysical and
epistemic modality (dualists, for short) hold that the phenomena
reflect a deep and fundamental distinction between two kinds of
modality. Monists, by contrast, believe that all the data can
ultimately be explained by appeal to a single kind of modality. They
may agree that there are cases in which a single sentence is, in some
sense, both necessary and a posteriori, or both contingent and a
priori. But they insist that there is no similar distinction at the
level of worlds or propositions. Rather, the phenomenon arises because
a single sentence can be associated with two different propositions,
one that is necessary and another that is contingent.

Dualists distinguish between two concepts of propositional
necessity, metaphysically necessity and epistemic
necessity. The two notions are not coextensive. At least some of
the sentences in Kripke's examples express propositions that
possess the one kind of necessity but not the
other.[4]

Once the existence of a distinctively metaphysical form of propositional necessity
is accepted, it is natural to wonder whether it is possible to say more
about its nature. Kit Fine (1994) offers an account of it that appeals
to the traditional distinction between those properties of a thing that
it possesses by its very nature and those that it has merely
accidentally. For example, it lies in the nature of water to be
composed of hydrogen and oxygen—being composed in this way is
part of what it is to be water—but it is merely accidental to
water that we use it to brush our teeth. A proposition is
metaphysically necessary just in case it is true in virtue of the
natures of things. Another account ties the metaphysical notion of
necessity constitutively to causation and explanation (Kment 2006a,b).

Dualism requires us to dismantle the picture of inquiry, information
and content sketched in the introduction to section 1. Note that it is
natural for a dualist to distinguish the space of
metaphysically possible worlds from the space of
epistemically possible worlds, i.e., from the space of
(maximally specific) ways the world might be that cannot be ruled out
on the basis of ideal reasoning alone, without empirical evidence
(Soames 2005, 2011). The range of epistemically possible worlds
outstrips the range of metaphysically possible worlds: there are some
ways the world couldn't have been, but which cannot be ruled by
ideal reasoning alone. For example, there is no metaphysically possible
world where gold has atomic number 78. But prior to carrying out the
right chemical investigations, we don't have enough evidence to
exclude all scenarios where gold has that atomic number, so some worlds
where gold has atomic number 78 are epistemically possible. Empirical
evidence is not used only to rule out (metaphysical) possibilities, but
is sometimes needed to rule out metaphysical impossibilities that are
epistemically possible. Consequently, we cannot in general identify
information with sets of metaphysically possible worlds, since we need
to distinguish between states of information in which the available
evidence rules out the same metaphysically possible worlds but
different metaphysically impossible worlds. By the same token, the
information encoded in a sentence cannot in general be identified with
a set of metaphysically possible worlds, since two sentences may be
true in all the same metaphysically possible worlds, but not in all the
same epistemically possible worlds. If we wanted to identify
information and sentential contents with sets of worlds, it would seem
more promising to use sets of epistemically possible worlds. But the
dualist may instead reject the possible-worlds account of information
and propositions altogether (see, e.g., Soames 1987, 2003, 395f.).

As mentioned above, monists explain the data described by Kripke by
holding that the sentences that figure in Kripke's examples are
associated with two different propositions, one that is necessary and
another that is contingent. This view comes in two main versions.
According to the first version, both propositions are semantically
expressed by the sentence. Proponents of this account need to formulate
a semantic theory that explains how that is possible. According to the
second version, only one of these propositions is semantically
expressed by the sentence, while the other is the proposition that is
communicated by a typical assertoric use of the sentence. A philosopher
holding this view needs to explain the pragmatic mechanism by which an
utterance of the sentence comes to communicate the second
proposition.

The first version of monism has been developed by David Chalmers and
Frank Jackson (Chalmers 1996, 1999, 2002a,b, 2004, 2006a,b; Chalmers and Jackson 2001; Jackson 1998, 2004, 2011), who build
on earlier work by David Kaplan (1989a,b), Gareth Evans (1979) and
Martin Davies and Lloyd Humberstone (1980), and others. On
Chalmers's and Jackson's view, what explains the phenomena
uncovered by Kripke is not a difference between two spaces of possible
worlds. There is only a single space of possible worlds: the
metaphysically possible worlds—the ways the world could
have been—just are the epistemically possible worlds: the
ways the world might be for all we can know independently of empirical
evidence. What explains the data is a difference between two different
ways in which sentences can be used to describe the worlds in
that space, i.e., between two different notions of a sentence's
being true in a world. The distinction can be illustrated by appeal to
our example of the proper name ‘Phosphorus.’ Suppose that
we have just introduced this name by using the description ‘the
last celestial body visible in the morning’ to fix its reference.
Consider a possible world w where the description singles out,
not Venus (as in our world), but Saturn. Assume further that in
w (as in the actual world), Venus is the second planet from
the sun, but Saturn is not. Consider:

(5)

Phosphorus is the second planet from the sun.

Is (5) true in w? There are two different ways of
understanding this question. On the one hand, it could mean something
roughly like this: if w actually obtains (contrary to what
astronomers tell us), is Phosphorus the second planet from the sun? The
answer to that question is surely ‘no.’
‘Phosphorus’ refers to whatever is actually the last
celestial body visible in the morning, and on the assumption that
w actually obtains, that object is Saturn, and is therefore
not the second planet. As Chalmers would put it, (5) is not true at
w considered as actual.[5]
But we can also interpret the question differently: if whad obtained, then would Phosphorus have been the
second planet? In considering that question, we are not hypothetically assuming that the object that actually satisfies the
reference-fixing description is Saturn. Instead, we can draw freely on
our belief that the object actually fitting the description is Venus,
so that the name picks out Venus in all possible worlds. Since Venus is
the second planet in w, it is true to say: if w had
obtained, then Phosphorus would have been the second planet. In
Chalmers's terminology, (5) is true at w considered as
counterfactual.

The distinction between the two concepts of truth in a world can be
explained within a theoretical framework known as two-dimensional
semantics, which assigns to a sentence like (5) an intension that
is a function, not from worlds to truth-values, but from pairs
of worlds to truth-values. The intension of (5) is the function that
assigns the true to a pair of worlds <u; w> just
in case the object that is the last celestial body visible in the
morning in u is the second planet in w.[6]
This account makes it easy to define
the two notions of truth in a world. A sentence P is true in
w considered as actual just in case the two-dimensional
function assigns the true to <w; w>.
P is true in w considered as counterfactual just in
case, where u is the actual world, the two-dimensional
function assigns the true to <u; w>.
Note that the two-dimensional intension of (5) determines whether (5)
is true at a world w considered as actual. But it does not in
general determine whether (5) is true at w considered as
counterfactual. That also depends on which world is actual. Knowledge
of a sentence's two-dimensional intension is therefore not in
general sufficient to know whether the sentence is true at w
considered as counterfactual. Further empirical evidence may be
required.

When combined with the conception of a sentence's content as
the set of worlds where it is true, the distinction between the two
concepts of truth in a world yields a distinction between two different
propositions expressed by a sentence. The first of these propositions
is the function that assigns the true to a world w just in
case the sentence is true in w considered as actual, while the
second proposition is the function that assigns the true to a world
w just in case the sentence is true in w considered
as counterfactual. Jackson calls the former proposition the
sentence's ‘A-intension’ (for ‘actual’)
and the latter its ‘C-intension’ (for
‘counterfactual’), while Chalmers calls the former the
‘primary intension’ and the latter the ‘secondary
intension.’ The distinction between the two propositions
expressed by a sentence yields a distinction between two notions of
sentential necessity: primary necessity, which applies to sentences
with necessary primary intensions, and secondary necessity, which
applies to sentences with necessary secondary intensions. If a sentence
has primary necessity, then that fact, and a fortiori the fact that the
sentence is true, can be read off its two-dimensional intension.
Therefore, if we know the two-dimensional intension, then that is
enough to know that the sentence is true. No further empirical
evidence is required. That motivates the thought that the notion of
primary necessity captures the idea of a prioricity or epistemic
necessity. The notion of secondary necessity, on the other hand, may be taken to capture the Kripkean idea of metaphysical necessity.

This account makes it straightforward to explain cases of a
posteriori necessity: they are simply cases of sentences whose
secondary intensions are necessary, but whose primary intensions are
contingent. Suppose that ‘Hesperus’ and
‘Phosphorus’ were introduced, respectively, by the
reference-fixing descriptions ‘the first celestial body visible
in the evening (if it exists)’ and ‘the last celestial body
visible in the morning (if it exists).’ Since the two
descriptions single out the same object in the actual world, the
sentence ‘If Hesperus exists, then Hesperus is Phosphorus’
is true in all worlds considered as counterfactual, and therefore has a
necessary secondary intension. However, in some non-actual worlds, the
two descriptions single out different objects. The sentence is false in
such a world considered as actual. The primary intension of the
sentence is therefore contingent.

An analogous account can be given of Kripke's examples of the
contingent a priori: these concern sentences whose primary intensions
are necessary and whose secondary intensions are contingent. Assume
again that the reference of ‘Julius’ is fixed by the
description ‘the inventor of the zip (if such a person
exists).’ Then in every world considered as actual, the name
singles out the person who is the inventor of the zip in that world (if
there is such a person) or nothing (if no such person exists in the
world). The primary intension of (4) is necessary. However, when we
evaluate (4) in a world w considered as counterfactual,
‘Julius’ picks out the individual who is the
actual inventor of the zip (provided that there actually is
such an individual and that he or she exists in w). And since
there are possible worlds where that individual exists but is not the
inventor of the zip, the secondary intension of (4) is contingent.

Chalmers (2002a, 2010) and Jackson (1998) have tried to support
their modal monism by arguing that it is gratuitous to postulate two
forms of modality, given that all the phenomena pointed out by Kripke
can be accommodated by appeal to a single kind of modality. Dualists
may reply that the greater simplicity in the view of modality has been
achieved only by adding complexity to the semantic theory. That
response could be answered by arguing that two-dimensional semantics
can be motivated by independent considerations. That, of course, is
controversial, as is the general viability of two-dimensional semantics
(see the entry
Two-Dimensional Semantics
for detailed discussion).

In addition, it is not obvious that the view of Chalmers and Jackson
can satisfactorily explain all the phenomena discussed in
section 1.1.
Some commentators have denied that it can give a viable general account
of Kripkean examples (see, e.g., Soames 2005; Vaidya 2008; Roca-Royes
2011). In any case, it is clear that the view can only explain how
necessity and epistemic necessity can come apart for sentences whose
primary and secondary intensions differ. That may be true of the cases
considered by Kripke, but it seems doubtful for the other examples
considered in section 1.1 (mathematical and metaphysical truths, laws,
and principles connecting the physical to the mental). In response,
Chalmers has argued that none of the latter cases are genuine examples
of the necessary a posteriori (1999, 2002a).

The second version of monism allows us to accommodate the
phenomena considered in section 1.1 while staying much closer to the
picture sketched in the introduction to section 1. On this view, the
data can be explained by appeal to a single space of possible worlds
and a single notion of truth in a world. The proposition semantically
expressed by a sentence containing a proper name or natural-kind term
is a function from individual worlds to truth-values. The proposition
expressed by ‘Phosphorus exists,’ e.g., is a function that
assigns the true to those worlds where Venus exists and the false to
the other worlds. (If the reference of ‘Phosphorus’ was
determined by a reference-fixing description together with the facts
about which entity meets the description, then that fact itself is not a semantic fact, but a metasemantic one, i.e., it does not concern the question of what the meaning of the word is, but the question of how the meaning of the word is determined.)
What explains the impression that a sentence like (1) expresses an a
posteriori claim is the fact that the proposition asserted by a typical
utterance of the sentence is not the one that is semantically expressed
by it, but a different proposition that is contingent and can only be
known empirically.

Robert Stalnaker (1978, 2001) has given a detailed account of the
pragmatic mechanism by which a contingent proposition comes to be
asserted by the utterance of a sentence that semantically expresses a
necessary proposition. On his account, linguistic communication evolves
in a context characterized by background assumptions that are shared
between the participants. These assumptions can be represented by the
set of worlds at which they are jointly true, which Stalnaker calls the
‘context set.’ The point of assertion is to add the
proposition asserted to the set of background assumptions and thereby
eliminate worlds where it is not true from the context set. To achieve
this, every assertion needs to conform to the rule that the proposition
asserted is false in some of the worlds that were in the context set
before the utterance (otherwise there are no worlds to eliminate) and
true in others (since the audience cannot eliminate all worlds from the
context set). Now consider a context where the shared background
assumptions include the proposition that the references of
‘A’ and ‘B’ were fixed by
certain descriptions but leave open whether the two descriptions single
out the same object. Suppose that someone says ‘A is
B.’ In every world in the context set, the sentence
semantically expresses either a necessary truth (if the two
descriptions single out the same object in the world) or a necessary
falsehood (if they don't). If the proposition that the speaker
intends to assert were the one that is semantically expressed by the
sentence, the aforementioned rule would be
violated.[7]
To avoid attributing this rule violation to the
speaker, the audience will construe the utterance as expressing a
different proposition, and the most natural candidate is the
proposition that the sentence uttered semantically expresses a true
proposition. (Stalnaker calls this the ‘diagonal
proposition.’) By exploiting this mechanism of reinterpretation,
a speaker can use the sentence to express the diagonal proposition.
This proposition is true in just those worlds in the context set where
the two descriptions single out the same object. It is clearly a
contingent proposition, and empirical evidence is required to know it.
Stalnaker suggests an analogous explanation of Kripke's proposed
cases of contingent a priori truth (1978, 83f.).

Stalnaker's account of the necessary a posteriori requires
that the proposition semantically expressed by the sentence and the
proposition that the sentence semantically expresses a truth hold in
different worlds in the context set. And that seems to require that the
assumptions shared between the participants of the conversation
don't determine what proposition is semantically expressed by the
sentence. It has been argued that that assumption is implausible in
some cases of Kripkean a posteriori necessities (Soames 2005, 96–105).
Suppose that I point to the desk in my office in broad daylight and say
‘That desk (if it exists) is made of wood.’ Unless the
context is highly unusual, the shared assumptions, so the argument
goes, uniquely determine what proposition is expressed by the
sentence.

It often seems very natural to use modal terminology when talking
about the laws of nature. We are inclined to say that nothing
can move faster than light to express the fact that the laws
rule out superluminal motion, and to state Newton's First Law by
saying that an object cannot depart from uniform rectilinear
motion unless acted on by an external force. This motivates the thought
that there is a form of necessity associated with the
natural
laws.[8]
It is controversial,
however, whether that form of necessity is simply metaphysical
necessity, or another kind of necessity. The former view is taken by
necessitarians (Swoyer 1982; Shoemaker 1980, 1998; Tweedale
1984; Fales 1993; Ellis 2001; Bird 2005), who believe that the laws (or
the laws conditionalized on the existence of the properties mentioned
in them) are metaphysically necessary. Contingentists deny
that, but many contingentists hold that there is a kind of necessity
distinct from metaphysical necessity that is characteristic of the laws
(e.g., Fine 2002), and which may be called natural or
nomic necessity. It is often assumed that nomic necessity is a
weaker form of necessity than metaphysical necessity: it attaches to
the laws and to all truths that are metaphysically necessitated by
them, so that anything that is metaphysically necessary is also
nomically necessary, but not vice versa.

Necessitarians have given several arguments for their position. Here
are two.

The argument from causal essentialism (e.g., Shoemaker
1980, 1998). Some philosophers believe that the causal powers that a
property confers on its instances are essential to it. Assuming that
causal laws describe the causal powers associated with properties, it
follows that these laws (or versions of them that are conditionalized
on the existence of the relevant properties) are necessary truths. This
is, in the first instance, only an argument for the necessity of
causal laws, but perhaps it can be argued that all laws of
nature are of this kind. Of course, even if this assumption is granted,
the argument is only as strong as the premise that properties have
their associated causal powers essentially. To support this view,
Sydney Shoemaker (1980) has given a battery of epistemological
arguments. He points out that our knowledge of the properties that an
object possesses can only rest on their effects on us, and must
therefore be grounded in the causal powers associated with these
properties. But, he goes on to argue that, without a necessary
connection between the properties and the associated causal powers, an
object's effects on us could not serve as a source of all the
knowledge about an object's properties that we take ourselves to
possess.

The argument from counterfactual robustness (Swoyer 1982;
Fales 1990, 1993; also see Lange 2004 for discussion). Natural laws are
often believed to differ from accidental generalizations by their
counterfactual robustness (counterfactual-supporting power).
If it is a law that all Fs are G, then this
generalization would still have been true if there had been more
Fs than there actually are, or if some
Fs had found themselves in conditions different from
the ones that actually obtain. For example, it would still have been
true that nothing moves faster than light if there had been more
objects than there actually are, or if some bodies had been moving in a
different direction. Contrast this with No emerald has ever
decorated a royal crown. That may be true, but it is not very
robust. It would have been false if some kings or queens of the past
had made different decisions. Some necessitarians have argued that
contingentism about the laws cannot provide a plausible explanation of
the special counterfactual robustness of the laws. Note that a
counterfactual “if it had been the case that
P, then it would have been the case that
Q” is usually taken to be true if Q
is true in those metaphysically possible P-worlds that are
closest to actuality. On this view, the special counterfactual
robustness of the law All Fs are G amounts, roughly
speaking, to this: of all the metaphysically possible worlds that
contain some additional Fs, or where some actual
Fs are in somewhat different circumstances, the ones
where the actual law holds are closer than the rest. If the laws hold
in some metaphysically possible worlds but not in others, then the
reason why the former are closer than the latter must be that the rules
we are using for deciding which worlds count as the closest say so. But
which such rules we use is a matter of convention. The
counterfactual-supporting power of the laws does not seem to be a
purely conventional matter, however. Necessitarianism, the argument
continues, offers a better explanation: the laws hold in the closest
possible worlds simply because they hold in all metaphysically
possible worlds. Conventions don't come into it. The
contingentist may reply that, even though the counterfactual robustness
of the laws is grounded in a convention, that convention may not be
arbitrary, but may have its rationale in certain features of
the laws that make them, in some sense, objectively important (Sidelle
2002), e.g., the fact that they relate to particularly pervasive and
conspicuous patterns in the history of the world.

Contingentism has often been defended by pointing out that the laws
of nature can be known only a posteriori, and that their negations are
conceivable (see Sidelle 2002). Necessitarians may reply to the first
point that Kripke's work has given us reasons for thinking that a
posteriori truths can be metaphysically necessary
(see section 1.1). In
response to the second point, they may grant that the negation of a law
is conceivable, but deny that conceivability is a good guide to
possibility (see the entry
Epistemology of Modality).
Alternatively,
they may deny that we can really conceive of a situation in which, say,
bodies violate the law of gravitation. What we can conceive of is a
situation in which objects move in ways that appear to violate
the law. But that situation cannot be correctly be described as
involving objects with mass. Rather, the objects in the
imagined situation have a different property that is very similar to
mass (call it ‘schmass’) but which is governed by slightly
different laws. Contingentists may reply that the non-existence of
schmass (or the non-existence of objects that move in the way imagined)
is itself a law, so that we have, after all, conceived of a situation
where one of the actual laws fails (see Fine 2002).

The concepts of metaphysical, epistemic, and nomic necessity are only
a few of the modal notions that figure in our thought and
discourse (as should be clear from the long list of uses of modal terms given in the introduction to this entry). We also speak of

and of a whole lot more. One would expect that some of these modal
concepts can be defined in terms of others. But how can that be done?
And is it possible to single out a small number of fundamental notions
of necessity in terms of which all the others can be defined?

It may be helpful in approaching these questions to distinguish
between two salient ways in which one modal property can be defined in
terms of another (Fine 2002, 254f.).

Restriction. To say that property N can be
defined from kind of necessity N* by restriction is
to say that a proposition's having N can be defined as
the combination of two things: (i) the proposition's having
N*, and (ii) its meeting certain additional conditions.

Relativization / quantifier restriction. To say that a
property N can be defined from a kind of necessity N*
by relativization to a class of propositions S is to
say that a proposition's having N can be defined as its
being N*-necessitated by S. A closely related way in
which a modal property can be defined in terms of another is by
quantifier restriction. Suppose that P* is a kind of
possibility that is the dual of N* (in the sense that it is
P*-possible that p just in case it's not
N*-necessary that not-p), and that we have at our
disposal the notion of a P*-possible world (a world that could
P*-possibly have been actualized). To say that the property
N can be defined from N* by quantifier restriction is
to say that that a proposition's having N can be defined
as its being true in all P*-possible worlds that meet a
certain condition C. (This is only the simplest way of
defining a modal property from a kind of necessity by quantifier
restriction. Much more sophisticated methods have been proposed. See,
e.g., Kratzer 1977, 1991.) Given reasonable assumptions, every
definition by relativization corresponds to a definition by quantifier
restriction, and vice
versa.[9]

Restriction allows us to define narrower modal properties from
broader ones. For example, it seems natural to hold that mathematical
necessity can be defined from metaphysical necessity by restriction.
(Perhaps a proposition's being mathematically necessary can be
defined as its being both metaphysically necessary and a mathematical
truth (Fine 2002, 255), or as its being metaphysically necessary
because it is a mathematical truth.) Relativization and
quantifier restriction, by contrast, allow us to define broader modal
properties in terms of narrower ones. For example, it may be held that
biological necessity can be defined as the property of being
metaphysically (or perhaps nomically) necessitated by the basic
principles of biology.

A modal property N is called alethic just in case
the claim that a proposition has N entails that the
proposition is true. Metaphysical, epistemic and nomic necessity are
all alethic. By contrast, moral and legal necessity are not. It is both
morally and legally necessary (i.e., it is required both by morality and
by the law) that no murders are committed, even though murders are in
fact being committed. A modal property defined by restriction from an
alethic kind of necessity must itself be alethic. By contrast,
relativization allows us to define non-alethic modal properties from
alethic ones, by relativizing to a class of propositions that contains
some falsehoods. Similarly, we can define a non-alethic modal property
from an alethic one by restricting the quantifier over possible worlds
to some class that does not include the actual world. For example,
legal necessity can perhaps be defined from metaphysical necessity by
restricting the quantifier to worlds where everybody conforms to the
actual laws.

The properties listed in (6) can very naturally be called
‘kinds of necessity,’ and in some contexts they are the
properties expressed by necessity operators like ‘must’ and
‘could not have been otherwise.’ But that is not true of
every property that can be defined from some kind of necessity by
relativization or restriction. For example, we can define a property by
relativizing metaphysical necessity to the class of truths stated in a
certain book, but it would not be natural at all to call this property
a kind of necessity. It is not plausible that there is a special form
of necessity that attaches to all and only the propositions
necessitated by the truths in the book. Similarly, the property defined
by restricting metaphysical necessity to the truths about cheddar
cheese cannot naturally be called a kind of necessity. There is no form
of necessity that applies to just those necessary propositions that
deal with cheddar and to none of the others. It is a good question what
distinguishes those properties defined by relativization and
restriction that we are willing to count as forms of necessity from the
rest. Perhaps the most natural answer is that the distinction is
dictated by our interests and concerns, and does not reflect a deep
metaphysical difference.

A more pressing question is whether some of the forms of necessity
discussed in sections 1
and 2 can be defined in terms of the others
by relativization or restriction. Consider epistemic and metaphysical
necessity first, and suppose for the sake of the argument that dualism
is true and the two properties are indeed different forms of
necessity. Can one of them be defined in terms of the other by one of
the aforementioned methods? Not if there are both necessary a
posteriori and contingent a priori propositions, since
relativization and restriction only allow us to define one property in
terms of another if the extension of one is a subclass of that of the
other. However, the existence of contingent a priori truths is more controversial than that of necessary a posteriori propositions, and someone trying to define epistemic necessity in terms of metaphysical necessity or vice versa may repudiate the contingent a priori and hold that the extension of epistemic necessity is included in that of metaphysical necessity. Then such a philosopher could try (a) to define metaphysical necessity from epistemic necessity by relativization to some suitable class, or (b) to define epistemic necessity from metaphysical necessity by restriction.

Such a definition may get the extension of the definiendum right.
But a definition may be intended to do much more than that: it may be
meant to tell us what it is for something to fall under the
concept to be defined. Suppose that someone tried to define the
property of being an equiangular triangle as that of being a triangle
whose sides are of equal length. While this is extensionally correct,
it does not give us the right account of what it is for something to be
an equiangular triangle (what it is for something to have that property
has something to do with the sizes of its angles, not with the lengths
of its sides). It could be argued that definitions of type (a) and (b)
face similar difficulties. For example, a definition of kind (a)
entails that a proposition's being metaphysically necessary
consists in its being epistemically necessitated by a certain class of
propositions. But that would make metaphysical necessity an epistemic
property, and dualists typically want to resist that idea. Similarly
for definitions of type (b). Whether something is epistemically
necessary (in the sense of being a priori) seems to be a purely
epistemic matter. A priori propositions may also be metaphysically
necessary, but their metaphysical necessity isn't part of what
makes them a priori, and therefore shouldn't be
mentioned in a definition of a prioricity.

If this argument is correct, then it is impossible to define
epistemic modal properties in terms of non-epistemic ones, or vice
versa. But what about metaphysical and nomic necessity? Suppose for the
sake of the argument that there is such a thing as nomic necessity (a form of necessity associated with the laws of nature) but that contingentism about the natural laws is true,
so that nomic necessity is indeed distinct from metaphysical necessity. Can we define one of these properties in terms of the other? The most natural way of doing this would be to say that

(7)

Nomic necessity can be
defined as the property of being metaphysically necessitated by the
laws of nature.

Such a definition may be extensionally accurate, and many
philosophers would not hesitate to endorse it. But others have doubted
that it captures what it is for a proposition to be nomically
necessary (Fine 2002). Nomic necessity is a special modal status
enjoyed by all and only the propositions that are metaphysically necessitated by the
natural laws. Now, if P is metaphysically necessitated by the
laws without itself being a law, then it may seem plausible to say, in
some sense, that P has that special modal status
because P is metaphysically necessitated by the laws.
But the reason why being metaphysically necessitated by the laws
confers that special modal status on P is presumably that the
laws themselves have that modal status and that this modal status gets
transmitted across metaphysical necessitation. But if we now ask what
makes it so that the laws themselves have that special modal status,
(7) does not seem to give us the correct answer: the special necessity
of the laws doesn't consist in the fact that they are
metaphysically necessitated by the laws. Hence, (7) cannot be a correct
general account of what constitutes that special modal status.

It is open to debate which kinds of necessity are fundamental, in
the sense that all others can be defined in terms of them, while they
are not themselves definable in terms of others. The monist view
considered in section 1.3, when combined with (7), may inspire the hope
that we can make do with a single fundamental kind of necessity. Others
have argued that there are several kinds of necessity that are not
mutually reducible. For example, Fine (2002) suggests (in a discussion
that sets aside epistemic modality) that there are three fundamental
kinds of necessity, which he calls ‘metaphysical,’
‘nomic’ and ‘normative’ necessity.

The reduction of the various kinds of necessity to a small number of
fundamental ones is an important step towards the goal of a unified
account of modality. But those who believe that there are several
different fundamental kinds of necessity need to address another
question: What is the common feature of these fundamental kinds of
necessity that makes them all kinds of necessity? Why do they count as
kinds of necessity, while other properties don't?

One strategy for answering this question, which centers on
non-epistemic forms of necessity, starts from a certain conception of
what (non-epistemic) necessity consists in: for a proposition to be
necessary is for its truth to be, in a certain sense, particularly
firm, secure, inexorable or unshakable in a wholly objective way. A
necessary truth could not easily have been false (it could less easily
have been false than a contingent truth). We may call this feature of a
proposition ‘modal force.’ It is natural to apply this
conception to metaphysical and nomic necessity. Each of these
properties may be held to consist in having a certain grade of modal
force, though if contingentism is true, the degree of modal force
required for nomic necessity is lower than that required for
metaphysical necessity. We could then say that a property is one of the
fundamental forms of necessity just in case a proposition
P's possessing that property consists entirely in
P's having a specific grade of modal force. Other kinds
of necessity, like those listed in (6) can be defined from the
fundamental ones by relativization or restriction. Having these
properties does not consist simply in having a specific grade
of modal force (and these properties therefore aren't among the fundamental kinds of necessity). For example, if a property is defined by relativizing
metaphysical necessity to a class of propositions S, then the
fact that a proposition P has that property consists in the
fact that the connection between S and P has
a certain grade of modal force. But that is not the same thing as P
itself having a certain grade of modal force. Similarly, if a
property is defined from, say, metaphysical necessity by restriction,
then having that property does not consist merely in
possessing such-and-such a grade of modal force, but in the conjunction
of that feature with some other property.

This approach evidently leaves the question how to understand the
idea of modal force (of a proposition's truth being very
unshakable). Some authors have attempted to explain this notion in
counterfactual terms (see Lewis 1973a, §2.5; Lewis 1973b, §2.1;
McFetridge 1990, 150ff.; Lange 1999, 2004, 2005; Williamson 2005, 2008;
Hill 2006; Kment 2006a): the necessary truths are distinguished from
the contingent ones by the fact that they are not only true as things
actually are, but that they would still have been true if things had
been different in various ways. To capture this idea more precisely,
Lange (2005) introduces the concept of ‘stability’: a
deductively closed set S of truths is stable just in case, for
any claim P in S and any claim Q consistent
with S, it is true in any context to say that it would still
have been the case that P if it had been the case that
Q. The different forms of necessity have in common that their
extensions are stable sets.

Kment (2006a) argues that modal force, and hence necessity and
possibility, come in many degrees. We often talk about such degrees of
possibility when we say things like ‘Team A could more easily
have won than Team B,’ ‘Team A could easily have won’
or ‘Team A almost won.’ The first utterance states that
A's winning had a greater degree of possibility than B's
winning, while the second and third simply ascribe a high degree of
possibility to A's winning. A proposition's degree of
possibility is the higher the less of a departure from actuality is
required for it to be true. Suppose, e.g., that Team A would have won
if one of their players had stood just an inch further to the left at a
crucial moment during the game. Then we can truly say that the team
could easily have won. More formally, P's degree of
possibility is the higher the closer the closest P-worlds are
to actuality. Similarly, a truth's degree of necessity is
measured by the distance from actuality to the closest worlds where it
is false. What metaphysical necessity, nomic necessity and the other
grades of necessity have in common is that each of them is the property
of having a degree of possibility that is above a certain threshold.
What distinguishes them is a difference in their associated
thresholds.

Hill, C. 2006, “Modality, Modal Epistemology, and the
Metaphysics of Consciousness,” in The Architecture of the
Imagination: New Essays on Pretense, Possibility, and Fiction,
S. Nichols (ed.), Oxford: Oxford University Press.

Hill, C. and B. McLaughlin, 1999, “There Are Fewer Things
in Reality Than Are Dreamt of in Chalmers's
Philosophy,” Philosophy and Phenomenological
Research, 59: 445–454.

Jackson, F., 1998, From Metaphysics to Ethics: A Defence of
Conceptual Analysis, Oxford: Oxford University Press.

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